Study of Miller IndicesMILLER INDICES

PROCEDURE FOR FINDING MILLER INDICES

DETERMINATION OF MILLER INDICES

IMPORTANT FEATURES OF MILLER INDICES

CRYSTAL DIRECTIONS

SEPARATION BETWEEN LATTICE PLANES

MILLER INDICES

The crystal lattice may be regarded as made

up of an infinite set of parallel equidistant

planes passing through the lattice points

which are known as lattice planes.

In simple terms, the planes passing through

lattice points are called ‘lattice planes’.

For a given lattice, the lattice planes can be

chosen in a different number of ways.

Miller Indices

MILLER INDICES

d

DIFFERENT LATTICE PLANES

Miller Indices

MILLER INDICES

The orientation of planes or faces in a crystal can be

described in terms of their intercepts on the three

axes.

Miller introduced a system to designate a plane in a

crystal.

He introduced a set of three numbers to specify a

plane in a crystal.

This set of three numbers is known as ‘Miller Indices’

of the concerned plane.

Miller Indices

MILLER INDICES

Miller indices is defined as the reciprocals of

the intercepts made by the plane on the three

axes.

Miller Indices

MILLER INDICES

Procedure for finding Miller Indices

Step 1: Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a,b and c.

Step 2: Determine the reciprocals of these numbers.

Miller Indices

Step 3: Find the least common denominator (lcd) and multiply each by this lcd.

Step 4:The result is written in paranthesis.This is called the `Miller Indices’ of the plane in the form (h k l).

This is called the `Miller Indices’ of the plane in the form (h k l).

MILLER INDICES

Miller Indices

ILLUSTRATION

PLANES IN A CRYSTAL

Plane ABC has intercepts of 2 units along X-axis, 3

units along Y-axis and 2 units along Z-axis.

Miller Indices

DETERMINATION OF ‘MILLER INDICES’

Step 1:The intercepts are 2,3 and 2 on the three axes.

Step 2:The reciprocals are 1/2, 1/3 and 1/2.

Step 3:The least common denominator is ‘6’. Multiplying each reciprocal by lcd, we get, 3,2 and 3.

Step 4:Hence Miller indices for the plane ABC is (3 2 3)

ILLUSTRATION

Miller Indices

IMPORTANT FEATURES OF MILLER INDICES

For the cubic crystal especially, the important features of Miller indices are,

A plane which is parallel to any one of the co-ordinate axes has an intercept of infinity (). Therefore the Miller index for that axis is zero; i.e. for an intercept at infinity, the corresponding index is zero.

MILLER INDICES

Miller Indices

EXAMPLE

( 1 0 0 ) plane

Plane parallel to Y and Z axes

Miller Indices

EXAMPLE

In the above plane, the intercept along X axis is 1 unit.

The plane is parallel to Y and Z axes. So, the intercepts along Y and Z axes are ‘’.

Now the intercepts are 1, and .

The reciprocals of the intercepts are = 1/1, 1/ and 1/.

Therefore the Miller indices for the above plane is (1 0 0).

Miller Indices

MILLER INDICES

IMPORTANT FEATURES OF MILLER INDICES

A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts.

All equally spaced parallel planes have same ‘Miller indices’ i.e. The Miller indices do not only define a particular plane but also a set of parallel planes. Thus the planes whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all represented by the same set of Miller indices.

Miller Indices

MILLER INDICES

IMPORTANT FEATURES OF MILLER INDICES

It is only the ratio of the indices which is important in this notation. The (6 2 2) planes are the same as (3 1 1) planes.

If a plane cuts an axis on the negative side of the origin, corresponding index is negative. It is represented by a bar, like (1 0 0). i.e. Miller indices (1 0 0) indicates that the plane has an intercept in the –ve X –axis.

Miller Indices

MILLER INDICES OF SOME IMPORTANT PLANES

Miller Indices

PROBLEMS

Worked Example:Calculate the miller indices for the plane with intercepts 2a, - 3b and 4c the along the crystallographic axes.

The intercepts are 2, - 3 and 4

Step 1: The intercepts are 2, -3 and 4 along the 3 axes

Step 2: The reciprocals are

Step 3: The least common denominator is 12.

Multiplying each reciprocal by lcd, we get 6 -4 and 3

Step 4: Hence the Miller indices for the plane is

1 1 1, and

2 3 4

6 4 3

Miller Indices

CRYSTAL DIRECTIONS

In crystal analysis, it is essential to indicate certain

directions inside the crystal.

A direction, in general may be represented in terms of

three axes with reference to the origin.In crystal system,

the line joining the origin and a lattice point represents

the direction of the lattice point.

Miller Indices

CRYSTAL DIRECTIONS

To find the Miller indices of a direction, Choose a perpendicular plane to that direction.

Find the Miller indices of that perpendicular plane.

The perpendicular plane and the direction have the same Miller indices value.

Therefore, the Miller indices of the perpendicular plane is written within a square bracket to represent the Miller indices of the direction like [ ].

Miller Indices

IMPORTANT DIRECTIONS IN CRYSTAL

Miller Indices

PROBLEMS

Worked ExampleFind the angle between the directions [2 1 1] and [1 1 2] in a

cubic crystal.

The two directions are [2 1 1] and [1 1 2]

We know that the angle between the two directions,

1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

½ ½

u u v v w wcos

(u v w ) (u v w )

Miller Indices

PROBLEMS

In this case, u1 = 2, v1 = 1, w1 = 1, u2 = 1, v2 = 1, w2 = 2

(or) cos = 0.833

= 35° 3530.

2 2 2 2 2 2

(2 1) (1 1) (1 2) 5cos

62 1 l 1 1 2

Miller Indices

DESIRABLE FEATURES OF MILLER INDICES

The angle ‘’ between any two crystallographic directions [u1 v1 w1] and [u2 v2 w2] can be calculated easily. The angle ‘’ is given by,

The direction [h k l] is perpendicular to the plane (h k l)

1 2 1 2 1 22 2 2 1/ 2 2 2 2 1/ 21 1 1 2 2 2

u u v v w wcos

(u v w ) (u v w )

Miller Indices

DESIRABLE FEATURES OF MILLER INDICES

The relation between the interplanar distance and the interatomic distance is given by,

for cubic crystal.

If (h k l) is the Miller indices of a crystal plane then the intercepts made by the plane with the crystallographic axes are given as

where a, b and c are the primitives.

2 2 2

ad

h k l

a b c, and

h k l

Miller Indices

SEPARATION BETWEEN LATTICE PLANES

Consider a cubic crystal of side ‘a’, and a

plane ABC.

This plane belongs to a family of planes

whose Miller indices are (h k l) because

Miller indices represent a set of planes.

Let ON =d, be the perpendicular distance of

the plane A B C from the origin.

Miller Indices

SEPARATION BETWEEN LATTICE PLANES

Miller Indices

SEPARATION BETWEEN LATTICE PLANES

Let 1, 1 and 1 (different from the interfacial angles, and ) be the angles between co- ordinate axes X,Y,Z and ON respectively.

The intercepts of the plane on the three axes are,

(1)a a aOA , OB and OC

h k l

Miller Indices

SEPARATION BETWEEN LATTICE PLANES

From the figure, 4.14(a), we have,

(2)

From the property of direction of cosines,

(3)

Using equation 1 in 2, we get,

1 1 11 1 1d d dcos ,cos and cos

OA OB OC

2 1 2 1 2 1cos cos cos 1

Miller Indices

SEPARATION BETWEEN LATTICE PLANES

Using equation 1 in 2, we get,

(4)

Substituting equation (4) in (3), we get,

1 1 11 1 1d h d k d lcos ,cos , and cos

a a a

2 2 2

1 1 1d h d k d l1

a a a

2 2 22 2 21 1 1

2 2 2

d h d k d l1

a a a

Miller Indices

i.e.

(5)

i.e. the perpendicular distance between the origin

and the 1st plane ABC is,

22 2 21

2

d(h k l ) 1

a

22

1 2 2 2

ad

(h k l )

1 2 2 2

ad ON

h k l

1 2 2 2

ad

h k l

Miller Indices

Now, let us consider the next parallel plane.

Let OM=d2 be the perpendicular distance of this plane from the origin.

The intercepts of this plane along the three axes are

1 1 12a 2a 2aOA ,OB ,OC ,

h k l

2 2 2 2

2aOM d

h k l

Miller Indices

• Therefore, the interplanar spacing between two adjacent parallel planes of Miller indices (h k l ) is given by, NM = OM – ON

i.e.Interplanar spacing

(6)

2 1 2 2 2

ad d d

h k l

SEPARATION BETWEEN LATTICE PLANES

Miller Indices

Worked Example

The lattice constant for a unit cell of aluminum is 4.031Å Calculate the interplanar space of (2 1 1) plane.

a = 4.031 Å(h k l) = (2 1 1)Interplanar spacing

d = 1.6456 Å

PROBLEMS

10

2 2 2 2 2 2

4.031 10ad

h k l 2 1 1

Miller Indices

PROBLEMS

Worked Example:Find the perpendicular distance between the two planes indicated by the Miller indices (1 2 1) and (2 1 2) in a unit cell of a cubic lattice

with a lattice constant parameter ‘a’.We know the perpendicular distance between the origin and the plane is (1 2 1)

and the perpendicular distance between the origin and the plane (2 1 2),

1 2 2 2 2 2 21 1 1

a a ad

6h k l 1 2 1

2 2 2 2 2 2 22 2 2

a a a ad

39h k l 2 1 2

Miller Indices

PROBLEMS

The perpendicular distance between the planes (1 2 1) and (2 1 2) are,

d = d1 – d2 =

(or) d = 0.0749 a.

3a 6a a(3 6)a a

36 3 6 3 6

Miller Indices